Gravitation: Potential: The Principle of Equivalence

Inertial and Gravitational Masses

The mass used in Newton's Second Law, = m_i is usually called the inertial mass. This mass is found with respect to a standard by measuring the respective acceleration of the mass and the standard when they are made to exert a force on one another. However, when two masses are weighed on a balance, the measurement records the gravitational force that is exerted by the earth on each mass that is measured. The mass determined in this way is called the gravitational mass and it is this mass that appears in Newton's Law of Universal Gravitation. The assertion that m_i = m_g is called the Principle of Equivalence.

There is no obvious reason why the inertial and gravitational masses should be equal. In fact, if two objects have inertial masses m₁ and m₂, and when tested by a balance are found to have equal weights w₁ and w₂, then:

w₁ = w₂âá’m₁g = m₂g

We can infer that m₁ = m₂ if and only if g is equal in both cases. That is, the principle of equivalence holds if the rate of fall due to gravity of different objects is identical. A great deal of experimental effort has been made to verifying this hypothesis. It has been determined that the equality holds to within one part in 10¹².

Einstein's Principle of Equivalence

Einstein's General Theory of Relativity is based on another principle of equivalence. This asserts that to a local observer (an observer inside the system), the effects experienced because of an acceleration are indistinguishable from the effects caused by a gravitational field. If an astronaut was trapped inside a spaceship with no window, and the spaceship was accelerating upwards at 9.8 m/sec², there is no experiment he could do to determine whether he was still on earth, or accelerating at a remote location in outer space.

Tides

In addition to the force of gravity from the earth, every object on the earth must necessarily feel a force from the moon and the sun. However, the earth is in free fall in relation to both these bodies. Just like the astronaut on the space shuttle discussed in Gravity Near the Earth the effects of the pull due to the sun and earth are "cancelled out" because of the free fall. Yet this cancellation is not exact; a small net force is exerted by both the moon and the sun on all objects on the earth. For objects fixed to the surface, this force is not significant. However, it does act on the oceans, causing them to bulge toward the moon (or sun) where the moon is closest to the earth and the force is strongest, and to bulge away where the force is weaker (on the opposite side from the moon).

Figure %: Effects of moon's gravity on earth's oceans.

As the earth rotates on its axis, the region facing the moon changes, causing the earth to shift slightly under the oceans. This effect accounts for the daily rise and fall of the tides.

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